Question

Without solving, how do you determine the number of solutions for the equation `sectheta=(2sqrt3)/3` in `-180^circ<=theta<=180^circ`?

Answer 1

Two solutions.
See explanation.

Explanation:

`sec theta = 1/(cos theta)=((2sqrt3)/3)`

So you're really looking at where
`cos(theta)=(3)/(2sqrt3)=(sqrt3)/2` (looks more familiar?)

`-180^@ = -pi` radian
`180^@ = pi` radian

Since the cosine function is "even", `cos (-x) = cos(x)` for all `x`.
So if there is a solution from 0 to `pi`, there is also a solution from 0 to `-pi`.

We can readily see that `0 < sqrt(3)/2 < 1`,
because `sqrt(3) < sqrt(4)`.
We know that the range of the cosine function is between -1 and 1, so this is good news, this means that there is a solution (if the value was larger than 1, we would have had zero solution).

Therefore,
`cos(theta)=sqrt(3)/2` has a solution that is between (0 and `180^@`), and another solution that is between (0 and `-180^@`).
Thus,
`sec theta= ((2sqrt3)/3)` has two solutions in the range `-180^@ < theta < 180^@`.
Q.E.D.